Question

Let $P\left(m\right)$ be the statement $m^2>100$, the statement $P(k+1)$ will be true if

• A. $P\left(1\right)$ is true
• B. $P\left(2\right)$ is true
• C. $P\left(k\right)$ is true
• D. none of these

Answer 1

Answer: (C)

$P\left(k\right)$ is true

$P\left(r\right)$ is true

$\Rightarrow r^2>100$ $\Rightarrow r^2+2r+1>100+2r+1$

$\Rightarrow {(r+1)}^2>100$

$\Rightarrow P(r+1)$ is true as

$r^2+(2r+1)>r^2>100$ $\Rightarrow P(k+1)$ is true (say $r=k$)

$P(k+1)$ is true when every $p\left(k\right)$ is

So, In order to prove that $P(k+1)$ is true.

It is sufficient to consider $P\left(k\right)$ is true.